Question: Bag A has 3 white marbles and 4 black marbles.  Bag B has 6 yellow marbles and 4 blue marbles.  Bag C has 2 yellow marbles and 5 blue marbles.  A marble is drawn at random from Bag A.  If it is white, a marble is drawn at random from Bag B, otherwise, if it is black, a marble is drawn at random from Bag C.  What is the probability that the second marble drawn is yellow?
We can get the second marble to be yellow in two ways:  either a white from A (with probability 3/7) then a yellow from B (with probability 6/10), or a black from A (with probability 4/7) then a yellow from C (with probability 2/7). Thus, the probability is \[ \left(\frac{3}{7}\times\frac{6}{10}\right)+\left(\frac{4}{7}\times\frac{2}{7}\right)=\boxed{\frac{103}{245}}.\]